“…13 As a result of self-images, the possibility of achieving a crossover of strip guides, a simple 3-dB directional coupler, and a filter ͑separating two wavelengths͒ was demonstrated by Ulrich and Ankele, 14 but only since the early 1990s has the concept been studied in more detail. Key papers written by Soldano, Penning, et al 4,6,15 analyzed the mathematics of the self-imaging phenomenon by calculating the coupling coefficients and predicting where multiple images can be found. Another key paper by Bachmann et al 16 in 1994 took this theory even further to calculate the phase relation between the input and the outputs, and outputs relative to each other.…”